The Multivariable Alexander Polynomial: Difference between revisions

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t[1] -> t1, t[2] -> t2, t[3] -> t4, t[4] -> t3
t[1] -> t1, t[2] -> t2, t[3] -> t4, t[4] -> t3
}</nowiki> |
}</nowiki> |
out= <nowiki>-t1 - t2 + t1 t2 - t3 + 2 t1 t3 + t2 t3 - t1 t2 t3 - t4 + t1 t4 +
out= <nowiki>-t1 - t2 t1 t2 - t3 2 t1 t3 t2 t3 - t1 t2 t3 - t4 t1 t4
2 t2 t4 - t1 t2 t4 + t3 t4 - t1 t3 t4 - t2 t3 t4</nowiki>}}
2 t2 t4 - t1 t2 t4 t3 t4 - t1 t3 t4 - t2 t3 t4</nowiki>}}
<!--END-->
<!--END-->


Line 44: Line 44:
n = 5 |
n = 5 |
in = <nowiki>mva - (mva /. {t1->t2, t2->t1})</nowiki> |
in = <nowiki>mva - (mva /. {t1->t2, t2->t1})</nowiki> |
out= <nowiki>t1 t3 - t2 t3 - t1 t4 + t2 t4</nowiki>}}
out= <nowiki>t1 t3 - t2 t3 - t1 t4 t2 t4</nowiki>}}
<!--END-->
<!--END-->


Line 55: Line 55:
There are 11 links with up to 11 crossings whose multivariable Alexander polynomial is <math>0</math>. Here they are:
There are 11 links with up to 11 crossings whose multivariable Alexander polynomial is <math>0</math>. Here they are:


<!--$$Select[AllLinks[], (MultivariableAlexander[#][t] == 0) &]$$-->
<!--$$Select[AllLinks[], (MultivariableAlexander[#][t] == 0)
<!--Robot Land, no human edits to "END"-->
{{InOut|
n = 6 |
in = <nowiki>Select[AllLinks[], (MultivariableAlexander[#][t] == 0) &]</nowiki> |
out= <nowiki>{Link[9, NonAlternating, 27], Link[10, NonAlternating, 32],
Link[10, NonAlternating, 36], Link[10, NonAlternating, 107],
Link[11, NonAlternating, 244], Link[11, NonAlternating, 247],
Link[11, NonAlternating, 334], Link[11, NonAlternating, 381],
Link[11, NonAlternating, 396], Link[11, NonAlternating, 404],
Link[11, NonAlternating, 406]}</nowiki>}}
<!--END-->

{{Knot Image Quadruple|L9n27|gif|L10n32|gif|L10n36|gif|L10n107|gif}}
{{Knot Image Quadruple|L11n244|gif|L11n247|gif|L11n334|gif|L11n381|gif}}
{{Knot Image Triple|L11n396|gif|L11n404|gif|L11n406|gif}}

[[User:Drorbn|Dror]] doesn't understand the multivariable Alexander polynomial well enough to give simple topological reasons for the vanishing of the said polynomial for these knots. (Though see the [[Talk:The Multivariable Alexander Polynomial|Talk Page]]).

====Detecting a Link Using the Multivariable Alexander Polynomial====

[[Image:Celtic-knot-basic-alternate.gif|thumb|right|200px|A mystery link]] On May 1, 2007 [[User:AnonMoos|AnonMoos]] asked [[User:Drorbn|Dror]] if he could identify the link in the figure on the right. So Dror typed:

<!--$$mva = MultivariableAlexander[L = PD[
X[1, 16, 2, 17], X[3, 15, 4, 14], X[5, 8, 6, 9],
X[7, 21, 8, 20], X[9, 22, 10, 13], X[11, 2, 12, 3],
X[13, 18, 14, 19], X[15, 12, 16, 1], X[17, 11, 18, 10],
X[19, 4, 20, 5], X[21, 7, 22, 6]
]][t]$$-->
<!--Robot Land, no human edits to "END"-->
{{InOut|
n = 7 |
in = <nowiki>mva = MultivariableAlexander[L = PD[
X[1, 16, 2, 17], X[3, 15, 4, 14], X[5, 8, 6, 9],
X[7, 21, 8, 20], X[9, 22, 10, 13], X[11, 2, 12, 3],
X[13, 18, 14, 19], X[15, 12, 16, 1], X[17, 11, 18, 10],
X[19, 4, 20, 5], X[21, 7, 22, 6]
]][t]</nowiki> |
out= <nowiki> 2 3 2
-1 + 3 t[1] - 3 t[1] + t[1] + 3 t[2] - 7 t[1] t[2] + 7 t[1] t[2] -
3 2 2 2 2
3 t[1] t[2] - 3 t[2] + 7 t[1] t[2] - 7 t[1] t[2] +
3 2 3 3 2 3 3 3
3 t[1] t[2] + t[2] - 3 t[1] t[2] + 3 t[1] t[2] - t[1] t[2]</nowiki>}}
<!--END-->

<!--$$Select[AllLinks[], (MultivariableAlexander[#][t] == mva) &]$$-->
<!--Robot Land, no human edits to "END"-->
{{InOut|
n = 8 |
in = <nowiki>Select[AllLinks[], (MultivariableAlexander[#][t] == mva) &]</nowiki> |
out= <nowiki>{Link[11, Alternating, 289]}</nowiki>}}
<!--END-->

And just to be sure,

<!--$${Jones[L][q], Jones[Link[11, Alternating, 289]][q]}$$-->
<!--Robot Land, no human edits to "END"-->
{{InOut|
n = 9 |
in = <nowiki>{Jones[L][q], Jones[Link[11, Alternating, 289]][q]}</nowiki> |
out= <nowiki> -(17/2) 4 8 12 16 18 17 15
{q - ----- + ----- - ----- + ---- - ---- + ---- - ---- +
15/2 13/2 11/2 9/2 7/2 5/2 3/2
q q q q q q q
10 3/2 5/2
------- - 7 Sqrt[q] + 3 q - q ,
Sqrt[q]
-(5/2) 3 7 3/2 5/2
-q + ---- - ------- + 10 Sqrt[q] - 15 q + 17 q -
3/2 Sqrt[q]
q
7/2 9/2 11/2 13/2 15/2 17/2
18 q + 16 q - 12 q + 8 q - 4 q + q }</nowiki>}}
<!--END-->

Thus the mystery link is [[L11a289]].

Revision as of 09:44, 18 June 2007


KnotTheory`/Navigation 

The Mathematica Package KnotTheory`

(For In[1] see Setup)

In[1]:= ?MultivariableAlexander
MultivariableAlexander[L][t] returns the multivariable Alexander polynomial of a link L as a function of the variable t[1], t[2], ..., t[c], where c is the number of components of L.
In[2]:= MultivariableAlexander::about
The multivariable Alexander program was written by Dan Carney at the University of Toronto in the summer of 2005.
L8a21.gif
L8a21

The link L8a21 is symmetric under cyclic permutations of its components but not under interchanging two adjacent components. It is amusing to see how this is reflected in its multivariable Alexander polynomial:

In[3]:= mva = MultivariableAlexander[Link[8, Alternating, 21]][t] /. { t[1] -> t1, t[2] -> t2, t[3] -> t4, t[4] -> t3 }
Out[3]= -t1 - t2 t1 t2 - t3 2 t1 t3 t2 t3 - t1 t2 t3 - t4 t1 t4 2 t2 t4 - t1 t2 t4 t3 t4 - t1 t3 t4 - t2 t3 t4
In[4]:= mva - (mva /. {t1->t2, t2->t3, t3->t4, t4->t1})
Out[4]= 0
In[5]:= mva - (mva /. {t1->t2, t2->t1})
Out[5]= t1 t3 - t2 t3 - t1 t4 t2 t4

But notice the funny labelling of the components! At the moment there is no way to tell MultivariableAlexander which variable is to be associated with what variable so MultivariableAlexander chooses an arbitrary ordering of tha variables. Hence we had to rename t[3] to be t4 and t[4] to be t3.

(To be precise, MultivariableAlexander orders the components so that its output would be lexicographically minimal among all possible orderings. This way it is at least guaranteed that different presentations for the same link will yield the same output for MultivariableAlexander.)

Links with Vanishing Multivariable Alexander Polynomial

There are 11 links with up to 11 crossings whose multivariable Alexander polynomial is . Here they are:


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